Divisibility Tricks for 2, 3, 6, and 9

An introduction to divisibility rules

In this article I will discuss how you can quickly tell if an integer is divisible by 2, 3, 6, or 9. There are also tricks for every other integer up to 12. The four tricks that follow are all closely related because the numbers 2, 3, 6, and 9 are very closely related.

First of all, let’s get a bit of vocabulary out in the open. In math, as in a few other aspects of life, it is somewhat important to understand the words that others are saying, typing, or yelling at you. Our first important piece of vocabulary is the word ‘integer’. The integers are the numbers 0, 1, 2, 3, and so on as well as -1, -2, -3, and on and on. Factors are integers that can be multiplied by other integers to produce a given number. For example, 2 and 3 are factors of 6 because 2 * 3 = 6. 2 and 3 are also factors of 12 because 2 * 6 = 12 and 3 * 4 = 12. Our last term is ‘divisible’ and it is closely related to factor. It’s easiest to understand as an example. 12 is divisible by 2 because 2 * 6 = 12. In other words, 12 is divisible by 2 because 2 is a factor of 12, and vice versa.

Now for our divisibility rules!

How to tell if an integer can be divided by 2!

Look at the last digit of the integer. If the last digit is 0, 2, 4, 6, or 8 then your number is divisible by 2. In other words your number is an even number.

Example:

12359370 can be divided by 2 because it ends in 0 and 25347 cannot be divided by 2 because it ends in 7. Congratulations, you have found an even number!

How to tell if an integer is divisible by 3!

Look at all of the digits in the number and add them up. Once you add them up you will have a new number. If 3 is a factor of that number then 3 is a factor of the original number. If you are not sure if the new number is divisible by 3 then you can make it even easier. Take all of the digits in the new number and add them up. If 3 is a factor of that number then 3 is a factor of the original number and the intermediate number. If you are still not sure if the newer number is divisible by 3 then you can make it even easier again. Just repeat the process until you get a number with which you are comfortable.

Examples:

51 has two digits, 5 and 1. 5 + 1 = 6 and 6 is divisible by 3 so 51 is divisible by 3.

85436 has 5 digits, 8 + 5 + 4 + 3 + 6 = 26 and I’m not sure if 26 is divisible by 3 (just kidding, I really do know) so I repeat the process again. 26 has two digits, 2 and 6. 2 + 6 = 8 and 8 is not divisible by 3 so 85436 is not divisible by 3. However, 85437 is divisible by 3 since 8 + 5 + 4 + 3 + 7 = 27, 2 + 7 = 9, and 9 is divisible by 3.

How to tell if an integer can be divided by 9!

You can use the same process as you did for checking if a number is divisible by 3. However, the process changes minimally. After you add up the digits you should now check if 9 is a factor of the new number. The reason that this process is so similar to the trick for deciding if 3 is a factor of a given number is because 9 = 3 * 3. This multiplicative relationship between 9 and 3 leads to many similarities between the two numbers.

Examples:

72 is divisible by 9 because 7 + 2 = 9 which is divisible by 9.

54678 is not divisible by 9 because 5 + 4 + 6 + 7 + 8 = 30 and 30 is not divisible by 9. You could repeat the process with 30 if you want. 3 + 0 = 3 and 3 is not divisible by 9, hence 54678 is not divisible by 9.

How to tell if an integer is divisible by 6!

Check if 2 is a factor of your given using the trick described above. Then, check if 3 is a factor of the number using the trick from above as well. If 2 and 3 are both factors, then 6 is also a factor of the given number. If it is not divisible by 2 and/or it is not divisible by 3, then it is not divisible by 6. The reason that you can apply the two aforementioned tricks is because 2 * 3 = 6.